Curvilinear boundary coordinates

We express for in the form where
is the outward normal coordinate, and
is a (periodic)
tangential location on the boundary , increasing in an anti-clockwise
fashion
(see Fig. I.1).
I will assume is a curve differentiable to sufficient order.
The corresponding locally-orthonormal unit vectors are (normal)
and (tangential).
In our curvilinear coordinates they are vector fields defined
by
and
.
For an introduction to such noncartesian systems see [7].
For the local components of the gradient of a scalar field
I will use the abbreviations
in the normal direction, and
in the tangential direction.
A general vector field will have the components with
and
.

Our task is to express (I.2) in terms of , ,
, and higher derivatives , etc.
We will choose a point
, so that these derivatives
are evaluated at the boundary.
The boundary conditions will later be applied.
We need rules for handling derivatives.
The metric is such that derivatives of the unit vectors obey

(I.3)

(I.4)

on the boundary ().
The local inverse radius of curvature of the boundary is

(I.5)

Thus gives the `connection' of the metric (or Christoffel symbol
[7]).
This means that while derivatives of scalars obey the usual Cartesian rules,
derivatives of vectors (or their local components) introduce extra terms.
This key observation was omitted in the work of VS [195,194].
For a general vector field we have,

(I.6)

(I.7)

(I.8)

(I.9)

The four derivative terms on the RHS are the components of the
covariant derivative tensor.
If we apply the above to the choice
which we take to be a constant
(translation) field according to the prescription of the previous section,
then all the derivatives of vanish leaving

(I.10)

(I.11)

(I.12)

which we will use later.
We will also need the divergence of a general
vector field , which is not
simply
, but rather

(I.13)

the trace of the covariant derivative tensor.
Choosing
finally gives the Laplacian